CSYMCambridge Suzuki Young Musicians (Cambridge, England, UK)
CSYMCyngor Sir Ynys Môn (Anglesey County Council, Wales, UK)
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Example 3.4 The graded Hopf algebras of ordered trees GSym, planar trees ySym, and divided powers CSym are all cofree, and so their compositions are cofree.
Our choice agrees with the product in ySym and CSym. In fact, the well-known Hopf algebra structures of ySym and CSym follow from their structure as graded Hopf operads.
Theorem 4.8 For C [member of] {GSym, ySym, CSym}, the coalgebra compositions C o CSym and CSym o C are connections on CSym.
Note that CSym o ySym is a connection on both CSym and on ySym, which gives two distinct one-sided Hopf algebra structures.
We describe the key definitions of Section 3.1 and Section 4 for CKSym := ySym o CSym. In the fundamental basis {[F.sub.p] | p [member of] CK*} of CKSym, the counit is [epsilon]([F.sub.p]) = [[delta].sub.0,[absolute value of p]] and the coproduct is
where g: CKSym [right arrow] CSym is the following connection.
1 is CSym o CSym, whose basis is indexed by combs over combs.